部分反馈下猜牌中的极值期望：Diaconis-Graham-Spiro 两个猜想的证明
Extremal expectations in card guessing with partial feedback: proofs of two conjectures of Diaconis-Graham-Spiro
- School of Data Science, Fudan University（复旦大学数据科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明 Diaconis-Graham-Spiro 关于部分反馈猜牌中最小与最大期望得分的两个猜想，给出精确界并揭示其渐近行为。
AI中文摘要：
我们研究在带是/否反馈的序贯猜牌中，所有自适应策略下期望得分的最大值与最小值。一副牌包含 $n$ 种标签，每种出现 $m$ 次，且均匀洗牌。牌被一张一张地猜测，每次猜测后，玩家仅被告知猜测是否正确。始终猜测同一标签恰好得到 $m$ 分。如果利用所有先前的反馈来寻求或避免正确猜测，期望得分能偏离 $m$ 多远？Diaconis、Graham 和 Spiro 猜想：当 $m\to\infty$ 且 $n$ 相对于 $m$ 足够大时，最小期望为 $m-o(m)$；并且，对于每个固定的 $m$，当 $n\to\infty$ 时最大期望的上极限至多为 $(e-1)m$。将这些极值记为 $P^-_{m,n}$ 和 $P^+_{m,n}$，我们证明 \\[ 0\le m-P^-_{m,n}\le3m^{3/4} \qquad(n\ge m\ge1), \\] 以及有限参数界 \\[ \frac{P^+_{m,n}}m\le\sum_{k=1}^{n}\frac1{k!}<e-1 \\] 对所有正整数 $m,n$ 成立。第一个界给出了对 $n\ge m$ 一致的一阶渐近；第二个界中的普适常数 $e-1$ 不能被减小。因此两个猜想均得证。下界结合了受限排列的切换比较与鞅二阶矩估计。对于上界，我们将重复的牌表示为独立的不同牌组的随机交错，并证明当组分牌组的标签被揭示时，最优期望是可加的。
英文摘要：
We study the minimum and maximum expected scores over all adaptive strategies in sequential card guessing with yes/no feedback. A deck contains $n$ labels, each appearing $m$ times, and is shuffled uniformly. The cards are guessed one at a time, and after each guess the player is told only whether it was correct. Always guessing the same label gives exactly $m$ points. How far can the expected score depart from $m$ if all previous feedback is used either to seek or to avoid correct guesses? Diaconis, Graham, and Spiro conjectured that the minimum expectation is $m-o(m)$ as $m\to\infty$ with $n$ sufficiently large in terms of $m$, and that, for each fixed $m$, the limit superior of the maximum expectation as $n\to\infty$ is at most $(e-1)m$. Writing these extrema as $P^-_{m,n}$ and $P^+_{m,n}$, we prove \[ 0\le m-P^-_{m,n}\le3m^{3/4} \qquad(n\ge m\ge1), \] and the finite-parameter bound \[ \frac{P^+_{m,n}}m\le\sum_{k=1}^{n}\frac1{k!}<e-1 \] for all positive integers $m,n$. The first bound gives a first-order asymptotic uniform in $n\ge m$; the universal constant $e-1$ in the second cannot be reduced. Thus both conjectures follow. The lower bound combines a switching comparison for restricted permutations with martingale second-moment estimates. For the upper bound, we represent the repeated cards as a random interleaving of independent decks of distinct cards and prove that the optimal expectation is additive when the component-deck labels are revealed.
